Non-relativistic spin splitting above and below the Fermi level in a $g$-wave altermagnet

Abstract: Altermagnets are distinguished by their unique spin group symmetries, where spin and spatial symmetries are fully decoupled, resulting in nonrelativistic spin splitting (NRSS) of electronic bands. This phenomenon, unlike conventional spin splitting driven by relativistic spin-orbit coupling, has transformative potential in fields such as spintronics, superconductivity and energy-efficient electronics. However, direct observation of NRSS is challenging due to presence of competing phases, low N\'eel temperatures, and the limitations of existing experimental probes to unambiguously capture the associated properties. Here, we integrate theoretical and experimental approaches to uncover NRSS in the intercalated transition metal dichalcogenide CoNb$_4$Se$_8$. Symmetry analysis, density functional theory (DFT), a novel Symmetry-Constrained Adaptive Basis (SCAB), and tight-binding modeling predict the presence of symmetry-enforced spin splitting, which we directly confirm using spin-ARPES for the occupied band structure and a newly developed technique, spin- and angle-resolved electron reflection spectroscopy (spin-ARRES), for the unoccupied states. Together, these complementary tools reveal alternating spin textures consistent with our predicted g-wave altermagnetic order and demonstrate the persistence of NRSS across a broad energy range. Crucially, temperature-dependent measurements show the suppression of NRSS at the N\'eel temperature ($T_N$), providing the first direct evidence of an altermagnetic phase transition. Residual spin splitting above $T_N$ suggests the coexistence of altermagnetic fluctuations and spin-orbit coupling effects, underscoring a complex interplay of mechanisms. By establishing CoNb$_4$Se$_8$ as a prototypical g-wave altermagnet, this work offers a robust framework for understanding NRSS, and lays the foundation for designing energy-efficient spin-based technologies.